Question

Note that whichever method you use, the expressions in terms of constants A,B, C etc. will always appear in numerators which are of at least 1 order of x lower than the order of the root in their respective denominators"

For which type of factors does the expression appear in the numerator with at least 1 order of x lower than the order of the root in their respective denominators?
a) Linear factors
b) Repeated factors
c) Quadratic irreducible factors
d) All of the above

Answer 1

Expressions have numerators of at least one order of x lower than the order of the root in their respective denominators when dealing with repeated factors because each term in the partial fraction decomposition corresponds to a power of the repeated factor with a lower-degree polynomial or constant as the numerator.

Step-by-step explanation:

The expressions with numerators of at least one order of x lower than the order of the root in their respective denominators are associated with b) Repeated factors. This occurs because when a polynomial is decomposed into partial fractions, each repeated linear factor gives rise to a series of terms in the decomposition. Each term corresponds to a power of the repeated factor, starting with the power of 1 and increasing by one until reaching the actual power of the factor in the denominator. However, the corresponding numerators only contain constants or polynomials of degree one less than the power of x in that term of the denominator. This rule holds similarly for repeated irreducible quadratic factors, where the numerators would be linear terms instead of just constants.

For linear factors not repeated, the numerator is usually just a constant. For a simple irreducible quadratic factor, the numerator will be a linear expression. However, for repeated factors, whether linear or quadratic, the degree of the numerator will always be one less than the corresponding power of the factor in the denominator.The expression appears in the numerator with at least 1 order of x lower than the order of the root in their respective denominators for all three types of factors: linear factors, repeated factors, and quadratic irreducible factors.For linear factors, the expression in the numerator will be one degree lower than the root. For example, if the root has an order of x^2, the expression in the numerator will have an order of x^1.Similarly, for repeated factors and quadratic irreducible factors, the expression in the numerator will be one order of x lower than the root in their respective denominators.